WAVE SPLITTING FOR FIRST-ORDER SYSTEMS OF EQUATIONS

Systems of first-order partial differential equations are considered and the possible decomposition of the solutions in forward and backward propagating is investigated. After a review of a customary procedure in the space-time domain (wave splitting), attention is addressed to systems in the Fourier-transform domain, thus considering frequencydependent functions of the space variable. The characterization is given for the direction of propagation and applications are developed to some cases of physical interest.


Introduction.
Wave propagation in inhomogeneous media and reflection and transmission processes are associated with various problems of mathematical interest.The governing equations are generally partial differential equations of various orders with variable coefficients.A basic question is whether and how the solution to the governing equations can be decomposed into forward-and backward-propagating waves.This question is of great interest in that the result of a scattering, or reflectiontransmission, process is required to consist of outgoing waves.According to the literature, the decomposition is often performed in the space-time domain through a wave-splitting technique [5].This technique works in homogeneous materials.Otherwise, the technique generates basis functions which though are not solutions to the governing equations.Also, even for homogeneous materials, the wave splitting is performed by carrying over to dispersive media (e.g., via convolutions) the propagation property of the principal, nondispersive part (see Section 2).
The governing equations are, for example, in the form (cf. [1]) in the unknown function u(z, t), z ∈ R, t ∈ R + , where c > 0 and a ≥ 0. The dependence of the coefficients a, b, c, d on the space variable z means that we allow the material system to be inhomogeneous.Otherwise, the model equations can be given the form of a first-order system where w is an n-tuple of unknown functions of z and t.The matrix C may have values in C n×n , with numerical entries possibly dependent on z, or may contain operators such that n-tuples of functions ∂ t w are mapped into n-tuples of functions ∂ z w.
To our mind, a more satisfactory approach is possible only within the Fouriertransform domain.In such a case, the solution to the Cauchy problem for monochromatic components is determined explicitly.Because of the inhomogeneity of the material, the solution is shown to result from a mixture of the components (mode conversion) thus making the wavesplitting in the space-time scarcely significant.For each monochromatic component, we characterize the forward or backward propagation through the phase function.Next, we apply the characterization to some cases of physical interest.They show that the direction of propagation may change with the frequency, which proves that a claimed direction of a solution in the space-time domain may cease to be true even in homogeneous materials.

Wave splitting in time domain.
Let n be an even integer, w : R → R n , A 0 ∈ R n×n , A : R + → R n×n .We assume that the pertinent equations can be written in the form This means that we are considering problems in the space-time variables z, t.
The wave splitting is accomplished by determining a new n-tuple of unknown functions such that the corresponding system involves a diagonal matrix.Let B 0 ∈ R n×n , B : R + → R n×n and consider w f ,w b ∈ R n/2 such that If ∂ t w f and ∂ t w b are continuous, then that is, the convolution Ꮾ and the derivative ∂ t commute.Now, observe that It is asserted that w f and w b are regarded as forward-and backward-propagating if ᏮᏭᏮ −1 is diagonal and the corresponding entry is appropriate.Upon solving suitable Volterra integral equations, the diagonal terms are determined, and, for example, the equation for w f takes the form (see [5, page 120]) While it is obvious that such w f is a forward-propagating wave if c is constant and ξ 0, we show later that an appropriate restriction on ξ is required so that the forward-propagating character is maintained.To gain this end, we find it convenient, if not imperative, to argue in the Fourier-transform domain.
3. First-order system in the Fourier-transform domain.Also, for specific applications, it is worth considering the equations of a dissipative material such as a viscoelastic body.We consider a one-dimensional viscoelastic solid along the z-axis.Let u be a transverse component of the displacement.The equation of motion takes the form where ρ is the mass density and τ is the traction component, namely, Here, µ 0 is the instantaneous shear modulus and µ , on R + , is the kernel characterizing the fading memory of the material.It is understood that ρ, µ 0 , and µ (•) may depend on z, which means that the body is allowed to be inhomogeneous.It is convenient to extend µ (z, •) to R by letting µ (z, ξ) = 0 as ξ < 0.
In terms of u and τ, we write the equations as Denote by the subscript F the Fourier transform Application of the Fourier transform yields Consequently, the pair w = [u, τ] satisfies the first-order system of ordinary equations where As a second case, we consider (1.1).Apply the Fourier transformation to obtain The pair w = [u, ∂ z u] then satisfies system (3.6)where Incidentally, upon Fourier transformation system, (1.2) takes the form (3.6) with A = iωC if C ∈ C n×n .Otherwise, if convolutions or time derivatives are involved in C, then A is determined by the Fourier transform properties.Anyway, we allow A in (3.6) to depend on z and be parametrized by the angular frequency ω.

Wave propagation in the Fourier-transform domain.
Assume that the problem under consideration takes the form (3.6) and that the matrix A is simple; namely, it has n linearly independent eigenvectors.Let λ 1 ,...,λ n be the eigenvalues, and p 1 ,...,p n ∈ C n the associated eigenvectors.For brevity, we omit specifying that each quantity is parameterized by ω.Moreover, let P be the matrix whose columns are the eigenvectors of A, that is, (4.1) Also, let Substitution of w F = P s in (3.6) provides where If the material is homogeneous, then ∂ z P = 0, and hence system (4.3)takes the diagonal form where Λ is independent of z.If, instead, the material is inhomogeneous (∂ z P ≠ 0), then in general, Q is a nondiagonal matrix and equations in (4.3) are not decoupled.Assume that A is independent of z, ∂ z P = 0, and hence (4.5) holds.If s is known at a value of z, say z = 0, we have give where To solve (4.8), we consider the propagator matrix Ω(z) such that It follows that where I is the n × n identity matrix.Hence, we have Upon the assumption that F is bounded, the solution Ω exists, is unique in L 2 (R), and is given by the Neumann series (cf.[2,6]) where the sequence of functions {F m } is defined by Consequently, we find that s(z where This in turn allows the original n-tuple w F to be written as (4.17) We now go back to the space-time domain through the inverse Fourier transform in the form where the dependences on ω are denoted explicitly.Result (4.18) provides the solution w(z, t) to system (3.6) with initial data on w(0,t).
Definition 5.1.The function f ω = ξ ω + iη ω of (5.1) represents a forward-(backward-) propagating wave if ω∂ z φ ω (z) < 0 (> 0). (5.8) It is natural to apply Definition 5.1 to plane (monochromatic) waves where (5.9) We have and hence φ ω (z) = ωz/c corresponds to a backward-propagating wave and φ ω (z) = −ωz/c corresponds to a forward-propagating wave.Now that the definition of wave propagating in one direction is available, we can investigate whether a pertinent field may be represented in terms of forward-and backward-propagating waves.

Applications.
We now look for the wave splitting in the Fourier-transform domain relative to particular systems of equations.

Dispersive media.
Apply the Fourier transform to (2.5) to have Hence, the eigenvalue implies that Hence, we have The wave is forward propagating if If, instead, ξ F < −1, then the wave is backward propagating.Incidentally, Consequently, we can have a forward-propagating ( ξ F > −1) wave while the sign of ∂ z α ω is unrestricted.This shows that the forward-propagating character is related to the condition ξ F > −1 of the kernel ξ and is not automatically induced by the principal part ∂ z + c −1 ∂ t of the operator in (2.5).

Dissipative wave equation.
For the sake of simplicity, we consider (1.1) with d, b = 0, and hence (3.6) holds with The eigenvalues λ of A satisfy Since a > 0, the number iωa − ω 2 c −2 is in the second quadrant.The eigenvalues λ 1 and λ 2 are then given by Hence, we have as λ = λ 1 (and the opposite as λ = λ 2 ).Since it follows that is a backward-propagating wave.This is consistent with the fact that and hence the amplitude decreases as z decreases as is natural of a backward-propagating wave.Of course, if a = 0, then and hence exp( propagates with a constant amplitude.In fact, the occurrence of and the dependence of c on z make w a z-dependent combination of the two elementary waves .16) 6.3.Shear waves in viscoelastic solids.System (3.6) holds with A as given by (3.7).Since µ = 0 on R − , we have where µ c and µ s are the half-range cosine and sine Fourier transforms of µ .Of course µ s (ω) is an odd function of ω.Also, by thermodynamics, we know that (cf.[3]) In addition, it is reasonable to assume that µ 0 + µ c (ω) > 0 for every ω ∈ R.
The eigenvalues λ satisfy where Hence, we have where Now, and then ω∂ z φ ω > 0 as ω > 0, ω∂ z φ ω < 0 as ω < 0. (6.27) Accordingly, as ω > 0, the wave associated with λ 1 is backward propagating and, as we expect it to be, exp α ω (z) increases as z increases.If, instead, ω < 0, then the wave is forward propagating and exp α ω (z) decreases as z increases.The opposite behaviour occurs with the wave associated with λ 2 .

Comments.
The dissipative wave equation associated with (6.7) allows us to establish a connection with the wave splitting technique (cf.[4]).Consider [4, (7)] with κ = 0 which coincides with (1.1) if b, d = 0, in the time domain, or (6.7) in the frequency domain.Via the operator K such that the functions are considered.It is not claimed that u + or u − are forward-or backward-propagating waves, but it is taken to be so if the medium is homogeneous as z < 0 or z > L. In such a case, the first-order system for u + ,u − decouples and takes the form where α is an operator involving a convolution and time derivatives.As shown for dispersive media modelled by (2.5), the decoupling of the first-order system does not guarantee that the pertinent function is forward or backward propagating.
In our approach, in the frequency domain, the homogeneity of the material implies that Ᏺ = 0 and that (4.18) reduces to Express w F (0,ω) as a linear combination of the two eigenvectors of A, w F (0,ω) = β 1 p 1 + β 2 p 2 .Hence, Since λ 1 (λ 2 ) is associated with a backward-(forward-) propagating wave, it follows that w(z, t) is the result of two waves in the time domain, namely w f being forward propagating and w b backward propagating.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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